Celebratio Mathematica

Donald L. Burkholder

Introduction to memorial issue
for Donald Burkholder (1927–2013)

by Rodrigo Bañuelos and Burgess Davis

Don Burk­hold­er had a long and dis­tin­guished ca­reer in math­em­at­ics. He is re­cog­nized world­wide for his deep and last­ing con­tri­bu­tions to mar­tin­gale the­ory and its ap­plic­a­tions to oth­er areas of math­em­at­ics. As a per­son, col­league and friend, he was kind and gen­er­ous, serving as ment­or and role mod­el for many young people en­ter­ing the field. Burk­hold­er pub­lished many of his found­a­tion­al pa­pers in IMS journ­als, par­tic­u­larly The An­nals of Math­em­at­ic­al Stat­ist­ics and The An­nals of Prob­ab­il­ity. His many con­tri­bu­tions to prob­ab­il­ity and ana­lys­is have been re­viewed in three re­cent pub­lic­a­tions: (1)  “Don Burk­hold­er’s work on Banach spaces,” (2)  “Don­ald Burk­hold­er’s work in mar­tin­gales and ana­lys­is,” and (3)  “The found­a­tion­al in­equal­it­ies of D. L. Burk­hold­er and some of their rami­fic­a­tions.” The first two art­icles, one writ­ten by Gilles Pis­i­er and the oth­er by Rodrigo Bañuelos and Bur­gess Dav­is, re­spect­ively, ap­peared in Se­lec­ted Works of Don­ald L. Burk­hold­er, pub­lished by Spring­er in 2010. The third art­icle, writ­ten by Bañuelos, ap­peared in the spe­cial volume Don Burk­hold­er: A col­lec­tion of art­icles in his hon­or, of the Illinois Journ­al of Math­em­at­ics, 2011. The read­er is re­ferred to these art­icles for an in-depth look at Burk­hold­er’s work, its ap­plic­a­tions and con­nec­tions to dif­fer­ent fields in math­em­at­ics where his work has had, and con­tin­ues to have, an im­pact. Here, we briefly touch upon a few of his pa­pers.

In 1966, Burk­hold­er pub­lished his cel­eb­rated “Mar­tin­gale trans­forms” pa­per in The An­nals of Math­em­at­ic­al Stat­ist­ics. In the 1930s, Mar­cinkiewicz and Pa­ley (in sep­ar­ate pa­pers) proved an in­equal­ity for the Haar sys­tem of func­tions in the unit in­ter­val, which is equi­val­ent to the bounded­ness of dy­ad­ic mar­tin­gale trans­forms with the pre­dict­able se­quence tak­ing val­ues in \( \{1, -1\}. \) Burk­hold­er’s pa­per ex­ten­ded this res­ult to the gen­er­al set­ting of mar­tin­gales. This pa­per was in­spired by the sem­in­al work of Don­ald Aus­tin, pub­lished in the same volume of The An­nals of Stat­ist­ics, which showed the fi­nite­ness of the square func­tion of \( L^1 \) bounded mar­tin­gales. The ground­break­ing pa­per, “Ex­tra­pol­a­tion and in­ter­pol­a­tion of quasi­lin­ear op­er­at­ors on mar­tin­gales,” writ­ten with Richard Gundy, was pub­lished in Acta Math­em­at­ica in 1970. The pa­per in­tro­duced the “good \( \lambda \) meth­od,” now im­port­ant in many areas of math­em­at­ics for com­par­ing norms of op­er­at­ors, and used it to prove the fam­ous Burk­hold­er–Gundy mar­tin­gale in­equal­it­ies which com­pare the norms of the square and max­im­al func­tions for all con­tinu­ous path mar­tin­gales and many oth­er reg­u­lar mar­tin­gales. In a sub­sequent pa­per Burk­hold­er, to­geth­er with Gundy and Mar­tin Sil­ver­stein, used these in­equal­it­ies to solve a long­stand­ing open prob­lem of Hardy and Lit­tle­wood con­cern­ing con­jug­ate func­tions. This re­volu­tion­ary pa­per in­spired many har­mon­ic ana­lysts to learn prob­ab­il­ity and many prob­ab­il­ists to learn har­mon­ic ana­lys­is, greatly en­rich­ing both fields.

For the next sev­er­al years, Burk­hold­er con­tin­ued to work on ap­plic­a­tions of mar­tin­gale the­ory and pro­duced many oth­er in­flu­en­tial pa­pers. His 1977 pub­lic­a­tion in Ad­vances in Math­em­at­ics, for ex­ample, ex­am­ines the exit times of Browni­an mo­tion from cones and oth­er do­mains in Eu­c­lidean spaces and ap­plies this to the be­ha­vi­or of har­mon­ic func­tions. This pa­per in­flu­enced many sub­sequent pa­pers on this and re­lated sub­jects.

In 1984 Burk­hold­er re­turned to bounded­ness of mar­tin­gale trans­forms. In his An­nals of Prob­ab­il­ity pa­per, “Bound­ary value prob­lems and sharp in­equal­it­ies for mar­tin­gale trans­forms,” he in­tro­duced the now called “Burk­hold­er meth­od.” This was a new ap­proach that re­proved his 1966 bounded­ness of mar­tin­gale trans­forms res­ults with op­tim­al con­stants and provided ex­ten­sions to the set­ting of Banach spaces. This al­lowed Burk­hold­er and oth­ers to an­swer many ques­tions about the geo­metry of Banach spaces for which both mar­tin­gale trans­forms and Calderón–Zyg­mund sin­gu­lar in­teg­ral op­er­at­ors (most not­ably the Hil­bert trans­form) act­ing on func­tions tak­ing val­ues in those Banach spaces, are bounded on \( L^p, \) \( 1 < p < \infty. \) The ques­tion of char­ac­ter­iz­ing the Banach spaces for which this prop­erty holds arose from work of Boch­ner and Taylor in the 1930s. The ideas in­tro­duced by Burk­hold­er in this pa­per were far ahead of their time and it took some 20 years for oth­ers to fully un­der­stand and ex­plore them. The ideas and tech­niques in the 1984 art­icle have been ex­tens­ively used in re­cent years on prob­lems in prob­ab­il­ity and har­mon­ic ana­lys­is where the tools of \( L^2 \) the­ory are either not avail­able or lead to es­tim­ates which are not op­tim­al. The wider im­plic­a­tions, in par­tic­u­lar its ap­plic­a­tion to sharp \( L^p \) bounds for sin­gu­lar in­teg­ral op­er­at­ors and rami­fic­a­tions in quasicon­form­al map­pings, prob­lems from the cal­cu­lus of vari­ations deal­ing with rank-one con­vex and quasicon­vex func­tions, prob­lems on op­tim­al con­trol and the the­ory of Bell­man func­tions, are all top­ics of cur­rent in­terest. One should note that these fields, on the sur­face, are far re­moved from mar­tin­gale the­ory.

Burk­hold­er served as pres­id­ent of the IMS from 1975 to 1976 and was ed­it­or of The An­nals of Math­em­at­ic­al Stat­ist­ics from 1964 to 1967. He served on mul­tiple ed­it­or­i­al boards and sci­entif­ic ad­vis­ory com­mit­tees for both the IMS and oth­er sci­entif­ic so­ci­et­ies. He su­per­vised 19 Ph.D. stu­dents and sev­er­al postdocs. He was elec­ted mem­ber of the Na­tion­al Academy of Sci­ences in 1992. He was a fel­low of the IMS, the Amer­ic­an Academy of Arts and Sci­ences, the So­ci­ety for In­dus­tri­al and Ap­plied Math­em­at­ics, the Amer­ic­an As­so­ci­ation for the Ad­vance­ment of Sci­ence and an in­aug­ur­al fel­low of the Amer­ic­an Math­em­at­ic­al So­ci­ety. In 1970 he was an in­vited speak­er at the ICM held in Nice, France.

Don Burk­hold­er grew up on a farm in Neb­raska dur­ing the years of the Great De­pres­sion and the Dust Bowl, which per­haps had something to do with his fe­ro­cious work eth­ic. After he re­tired from his dis­tin­guished pro­fess­or­ship po­s­i­tion at the Uni­versity of Illinois he was asked if he was now tak­ing it easy, to which he replied “I am, I no longer work on Sunday af­ter­noons.” His char­ac­ter also showed in how kind and help­ful he was to oth­ers and spe­cially to young math­em­aticians. He was al­ways en­cour­aging and nev­er in­tim­id­at­ing.

Burk­hold­er at­ten­ded Earl­ham Col­lege where he met Jean, his wife and the moth­er of their three chil­dren. He re­ceived his Ph.D. in math­em­at­ic­al stat­ist­ics from the Uni­versity of North Car­o­lina at Chapel Hill in 1955 where he worked un­der the math­em­at­ic­al stat­ist­i­cian Wassily Hoeff­d­ing, one of the founders of non­para­met­ric stat­ist­ics. His thes­is, “On a cer­tain class of stochast­ic ap­prox­im­a­tion pro­cesses,” pub­lished in The An­nals of Math­em­at­ic­al Stat­ist­ics, ex­ten­ded work of Ju­li­us Blum, Kai Lai Chung, Joseph Hodges, and Erich Leo Lehmann on “asymp­tot­ic nor­mal­ity.” Im­me­di­ately after gradu­at­ing he took a job at the Uni­versity of Illinois at Urb­ana-Cham­paign where the le­gendary Joseph L. Doob, work­ing on mar­tin­gales and ap­plic­a­tions of prob­ab­il­ity to po­ten­tial the­ory and com­plex ana­lys­is, was a pro­fess­or. Al­most cer­tainly in­flu­enced by Doob, Burk­hold­er soon turned his at­ten­tion to these sub­jects.

Dur­ing his en­tire ca­reer, Burk­hold­er wrote only six joint re­search pa­pers. His closest col­lab­or­at­or, and the only math­em­atician with whom he wrote more than one pa­per, was Richard Gundy. Their work had a deep and last­ing im­pact in prob­ab­il­ity and its ap­plic­a­tions. Burk­hold­er’s later work in the eighties on mar­tin­gales tak­ing val­ues in Banach spaces led him to new tech­niques to prove sharp mar­tin­gale in­equal­it­ies. The lat­ter have had ap­plic­a­tions in areas which on the sur­face are far re­moved from mar­tin­gales. The wide use of Burk­hold­er’s ideas, res­ults and tech­niques, shows the uni­ver­sal­ity of his math­em­at­ics, al­though he seem­ingly “only” touched mar­tin­gales.